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We construct a unified operator framework for quantum holonomies generatedfrom bosonic systems. For a system whose Hamiltonian is bilinear in thecreation and annihilation operators, we find a holonomy group determined onlyby a set of selected orthonormal modes obeying a stronger version of theadiabatic theorem. This photon-number independent description offers deeperinsight as well as a computational advantage when compared to the standardformalism on geometric phases. In particular, a strong analogy between quantumholonomies and linear optical networks can be drawn. This relation provides anexplicit recipe how any linear optical quantum computation can be madegeometrically robust in terms of adiabatic or nonadiabatic geometric phases.

Geometrically robust linear optics from non-Abelian geometric phases